Optimal Mean Reversion Trading: Mathematical Analysis And Practical Applications

Mathematical Analysis and Practical Applications

Tim Leung, Xin Li

• Publisher: WSPC
• Year: 2015
• eBook ISBN: 9789814725934
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Optimal Mean Reversion Trading: Mathematical Analysis and Practical Applications provides a systematic study to the practical problem of optimal trading in the presence of mean-reverting price dynamics. It is self-contained and organized in its presentation, and provides rigorous mathematical analysis as well as computational methods for trading ETFs, options, futures on commodities or volatility indices, and credit risk derivatives.

This book offers a unique financial engineering approach that combines novel analytical methodologies and applications to a wide array of real-world examples. It extracts the mathematical problems from various trading approaches and scenarios, but also addresses the practical aspects of trading problems, such as model estimation, risk premium, risk constraints, and transaction costs. The explanations in the book are detailed enough to capture the interest of the curious student or researcher, and complete enough to give the necessary background material for further exploration into the subject and related literature.

This book will be a useful tool for anyone interested in financial engineering, particularly algorithmic trading and commodity trading, and would like to understand the mathematically optimal strategies in different market environments.

Optimal Mean Reversion Trading: Mathematical Analysis and Practical Applications provides a systematic study to the practical problem of optimal trading in the presence of mean-reverting price dynamics. It is self-contained and organized in its presentation, and provides rigorous mathematical analysis as well as computational methods for trading ETFs, options, futures on commodities or volatility indices, and credit risk derivatives.

This book offers a unique financial engineering approach that combines novel analytical methodologies and applications to a wide array of real-world examples. It extracts the mathematical problems from various trading approaches and scenarios, but also addresses the practical aspects of trading problems, such as model estimation, risk premium, risk constraints, and transaction costs. The explanations in the book are detailed enough to capture the interest of the curious student or researcher, and complete enough to give the necessary background material for further exploration into the subject and related literature.

This book will be a useful tool for anyone interested in financial engineering, particularly algorithmic trading and commodity trading, and would like to understand the mathematically optimal strategies in different market environments.

Readership: Doctoral and master's students, advanced undergraduates, practitioners, and researchers in financial engineering, with a particular interest or specialization in algorithmic trading (especially pairs trading) and ETFs, futures, commodities, volatility derivatives and credit risk.
Key Features:

• Contains both an analysis of trading strategies and methods and means of practical implementation
• Approaches the topic using a balanced approach of rigorous analysis and real-world examples taken from a variety of market sectors such as fixed income funds, commodities, index/volatility futures, and options
• Includes detailed analysis of ETF-based pairs trading strategies, and other mean reversion strategies
• Explains issues involved in the day-to-day life of traders, going beyond the mathematics of trading
• Provides mathematical justification and quantitative enhancement for certain intuitive trading strategies that can be used by practitioners

Chapter 1

Introduction

In the financial markets, it has been widely observed that many asset prices exhibit mean reversion, including commodities, foreign exchange rates, volatility indices, as well as US and global equities.¹ Mean-reverting processes are also used to model the dynamics of bond prices, interest rate, and default risk. In order to visualize a mean-reverting price path, we illustrate in Figure 1.1(a) the historical prices of an exchange-traded fund (ETF), the Vanguard Short-Term Bond ETF (BSV), from June 12, 2014 to June 11, 2015. This ETF is designed to track bond prices with short maturities, and is traded liquidly on NYSE and other exchanges. As another example, Figure 1.1(b) shows the time series of CBOE Volatility Index (VIX) from June 12, 2014 to June 11, 2015. Although the volatility index is not traded, investors can gain exposure to it by trading futures, options, or exchange-traded notes (ETNs) designed to track the index.²

In industry, hedge fund managers and investors often attempt to construct mean-reverting prices by simultaneously taking positions in two highly correlated or co-moving assets. The advent of exchange-traded funds has further facilitated this pairs trading approach since some ETFs are designed to track identical or similar indexes and assets. Empirical studies have found that the spreads between commodity ETFs, such as physical gold and gold equity ETFs, are mean-reverting, and such price behavior has been used for statistical arbitrage.³

Fig. 1.1 Historical price paths of (a) Vanguard Short-Term Bond ETF (BSV) and (b) CBOE Volatility Index (VIX), respectively, from June 12, 2014 to June 11, 2015.

On the other hand, one important problem commonly faced by individual and institutional investors is to determine when to open and close a position. While observing the prevailing market prices, a speculative investor can choose to enter the market immediately or wait for a future opportunity. After completing the first trade, the investor will need to decide when is the best to close the position. This motivates the investigation of the optimal sequential timing of trades.

Naturally, the optimal sequence of trading times depend on the price dynamics of the risky asset. For instance, if the price process is a super/submartingale, that is, decreasing/increasing on average, then the investor who seeks to maximize the expected liquidation value will either sell immediately or wait forever. Such a trivial timing arises when the underlying price follows a geometric Brownian motion (see Example 2.1 below).

In this book, we study the optimal timing of trades for assets or portfolios that have mean-reverting dynamics. Specifically, we provide detailed mathematical analysis and implementation methods for various trading problems mainly under three important mean-reverting models: Ornstein-Uhlenbeck (OU), exponential Ornstein-Uhlenbeck (XOU), and Cox-Ingersoll-Ross (CIR) models. Due to their tractability and interpretability, these models are widely used in practice for describing and estimating mean reversion in asset prices. Therefore, the objective of this book is to introduce optimality criteria and discuss solution methods for an array of trading problems, and we focus on developing optimal strategies that maximize expected returns with controlled/limited risks.

1.1 Chapter Outline

In Chapter 2, we study the optimal timing of trades subject to transaction costs under the OU model. We motivate through a pairs trading example where the resulting optimized portfolio value admits an OU process. The trading strategies are implemented for the application of trading a pair of ETFs with similar underlying assets. Mathematically, our formulation leads to an optimal double stopping problem that gives the optimal entry and exit decision rules. We obtain analytic solutions for both the entry and exit problems. In addition, we incorporate a stop-loss constraint to our trading problem. We find that a higher stop-loss level induces the investor to voluntarily liquidate earlier at a lower take-profit level. Moreover, the entry region is characterized by a bounded price interval that lies strictly above stop-loss level. In other words, it is optimal to wait if the current price is too high or too close to the lower stop-loss level. This is intuitive since entering the market close to stop-loss implies a high chance of exiting at a loss afterwards. As a result, the delay region (complement of the entry region) is disconnected. Furthermore, we show that optimal liquidation level decreases with the stop-loss level until they coincide, in which case immediate liquidation is optimal at all price levels.

To incorporate mean-reversion for positive price processes, one popular choice for pricing and investment applications is the exponential OU model, as proposed by Schwartz (1997) for commodity prices, due to its analytical tractability. It also serves as the building block of more sophisticated mean-reverting models. In Chapter 3, we study the optimal timing of trades under the XOU model. We consider the optimal double stopping problem, as well as a different but related formulation. In the second formulation, the investor is assumed to enter and exit the market infinitely many times with transaction costs. This gives rise to an optimal switching problem. We analytically derive the non-trivial entry and exit timing strategies. Under both approaches, it is optimal to sell when the asset price is sufficiently high, though at different levels. As for entry timing, we find that, under some conditions, it is optimal for the investor not to enter the market at all when facing the optimal switching problem. In this case for the investor who has a long position, the optimal switching problem reduces into an optimal stopping problem, where the optimal liquidation level is identical to that of the optimal double stopping problem. Otherwise, the optimal entry timing strategies for the double stopping and switching problem are described by the underlying’s first passage time to an interval that lies above level zero. In other words, the continuation region for entry is disconnected of the form (0, A) ∪ (B,+∞), with critical price levels A and B (see Theorems 3.4 and 3.7 below). This means that the investor generally enters when the price is low, but may find it optimal to wait if the current price is too close to zero. We find that this phenomenon is a distinct consequence due to fixed transaction costs under the XOU model.

In Chapter 4, we turn to the trading problems when the asset follows the CIR process. The CIR process has been widely used as the model for interest rate, volatility, commodity, and energy prices.⁴ The main focus of the chapter is the analytical derivation of the non-trivial optimal entry and exit timing strategies and the associated value functions. Under both double stopping and switching approaches, it is optimal to exit when the process value is sufficiently high, though at different levels. As for entry timing, we find the necessary and sufficient conditions whereby it is optimal not to enter at all when facing the optimal switching problem. In this case, the optimal switching problem in fact reduces to an optimal single stopping problem, where the optimal stopping level is identical to that of the optimal double stopping problem.

A typical solution approach for optimal stopping problems driven by diffusion involves the analytical and numerical studies of the associated free boundary problems or variational inequalities (VIs); see, for example, Bensoussan and Lions (1982), Øksendal (2003), and Sun (1992). For our double optimal stopping problem, this method would determine the value functions from a pair of VIs and require regularity conditions to guarantee that the solutions to the VIs indeed correspond to the optimal stopping problems. As noted by Dayanik (2008), “the variational methods become challenging when the form of the reward function and/or the dynamics of the diffusion obscure the shape of the optimal continuation region.” In our optimal entry timing problem, the reward function involves the value function from the exit timing problem, which is not monotone and can be positive and negative.

In contrast to the variational inequality approach, our proposed methodology for Chapters 2 to 4 applies probabilistic arguments to analytically characterize the optimal stopping value functions as the smallest concave majorant of the corresponding reward function. A key feature of this approach is that it allows us to directly construct the value function, without a priori finding a candidate value function or imposing conditions on the stopping and delay (continuation) regions, such as whether they are connected or not. In other words, our method will derive the structure of the stopping and delay regions as an output. Having solved the optimal double stopping problem, we determine the optimal structures of the buy/sell/wait regions. We then apply this to infer a similar solution structure for the optimal switching problem and verify using the variational inequalities.

Chapters 5 to 7 are dedicated to trading of financial derivatives, namely, futures, options, and credit derivatives, respectively. Started as contracts for the delivery of agricultural products decades ago, futures are now one of the most common form of financial derivatives, and there are a high number of tradable commodities, including agricultural products, livestocks, precious metals, oil and gas, as well as other underlyings such as interest rates, currency, equity and volatility indices. Each futures contract stipulates the buyer to purchase (seller to sell) a fixed quantity of a commodity at a fixed price to be paid for on a pre-specified future date. Many futures require physical delivery of the commodity, but some, like the VIX futures, are settled in cash. In Chapter 5, we discuss the pricing of futures, explore the timing options embedded in futures trading, and develop optimal dynamic speculative strategies for market entry and exit. Focusing on the applications to commodity and volatility futures, we analyze these problems under mean-reverting spot price dynamics.

For decades, options have been widely used as a tool for investment and risk management. As of 2012, the daily market notional for S&P 500 options is more than US$90 billion and the average daily volume has grown rapidly from 119,808 in 2002 to 839,108 as of Jan 2013.⁵ Empirical studies on options returns often assume that the options are held to maturity (see Broadie et al. (2009) and references therein). In a liquid options market, such as the S&P 500 index, VIX, or gold options markets, there is an intrinsic timing flexibility to liquidate the position through the market prior to expiry. This leads us to investigate the optimal time to liquidate an option position. In Chapter 6, we propose a risk-adjusted optimal stopping framework to address this problem for a variety of options under different underlying price dynamics.

In addition to maximizing the expected discounted market value to be received from option sale, we incorporate a risk penalty that accounts for adverse price movements till the liquidation time. Specifically, we measure the associated risk in terms of the realized shortfall of the option position, and thus introduce the trade-off between risk and return for every liquidation timing strategy. Under a general diffusion model for the underlying stock price, we formulate an optimal stopping problem that includes an integral penalization term. To this end, we define and apply the concept of optimal liquidation premium which represents the additional value from optimally waiting to sell, as opposed to immediate liquidation. As it turns out, it is optimal for the option holder to sell as soon as this premium vanishes. This observation leads to a number of useful mathematical characterizations and financial interpretations of the optimal liquidation strategies for various positions.

Lastly, in Chapter 7 we propose a new approach to tackle the optimal liquidation problem for credit derivatives. The first step is to understand how the market compensates investors for bearing credit risk. We examine analytically the structure of default risk premia inferred from the market prices of corporate bonds, credit default swaps, and multi-name credit derivatives. We identify the risk premium components, namely, the mark-to-market risk premium that accounts for the fluctuations in default risk, as well as the event risk premium (or jump-to-default risk premium) that compensates for the uncertain timing of the default event. Our approach is to first provide a general mathematical framework for price discrepancy between the market and investors under an intensity-based credit risk model. Then, we derive and analyze the optimal stopping problem corresponding to the liquidation of credit derivatives under price discrepancy.

In order to measure the benefit of optimally timing to sell as opposed to immediate liquidation, we define and quantify the so-called delayed liquidation premium. We analyze the scenarios where immediate or delayed liquidation is optimal. Moreover, through its probabilistic representation, the delayed liquidation premium reveals the roles of risk premia in the liquidation timing. We investigate the investor’s liquidation timing for various credit derivatives, including defaultable bonds, CDSs, as well as, multi-name credit derivatives, in markets where the default intensity and interest rate processes are mean-reverting. The impact of price discrepancy is revealed through a series of numerical examples illustrating the trader’s optimal liquidation strategies.

1.2 Related Studies

In the context of pairs trading, a number of studies have also considered market timing strategy with two price levels. For example, Gatev et al. (2006) study the historical returns from the buy-low-sell-high strategy where the entry/exit levels are set as ±1 standard deviation from the long-run mean. Similarly, Avellaneda and Lee (2010) consider starting and ending a pairs trade based on the spread’s distance from its mean. In Elliott et al. (2005), the market entry timing is modeled by the first passage time of an OU process, followed by an exit at a fixed finite horizon. In comparison, rather than assigning ad hoc price levels or fixed trading times, our approach in Chapter 2, adapted from Leung and Li (2015), generates the entry and exit thresholds as solutions of an optimal double stopping problem. Considering an exponential OU asset price with zero log mean, Bertram (2010) numerically computes the optimal enter and exit levels that maximize the expected return per unit time. Other timing strategies adopted by practitioners have been discussed in Vidyamurthy (2004). Song et al. (2009) and Song and Zhang (2013) study the optimal switching problem with stop-loss under the OU price dynamics. In their recent book, Cartea et al. (2015) also study the pairs trading problem that allows the investor to enter the market by longing one asset and shorting the other, or taking the opposite position, thus resulting in a two-sided market entry strategy.

In Chapters 3 and 4, we consider optimal double stopping and optimal switching problems under exponential OU and CIR models with fixed transaction costs. These two chapters are based on Leung et al. (2015) and Leung et al. (2014). In particular, the optimal entry timing with fixed transaction costs is characteristically different from that with slippage (see Czichowsky...

Table of contents

• Cover page
• Title page
• Copyright page
• Preface
• Contents
• 1. Introduction
• 2. Trading Under the Ornstein-Uhlenbeck Model
• 3. Trading Under the Exponential OU Model
• 4. Trading Under the CIR Model
• 5. Futures Trading Under Mean Reversion
• 6. Optimal Liquidation of Options
• 7. Trading Credit Derivatives
• Bibliography
• Index